Reaction network realization of nonnegative delayed dynamical models

2019. 04. 25. 12:00
Gábor Szederkényi, co-authors: Mihály Vághy, Gergely Szlobodnyik

Delayed processes are ubiquitous in nature and technology, and it is often required to explicitly include time delay into dynamical models to describe the studied phenomena in a satisfactory way. The need for using delayed reactions in kinetic models appeared in the literature already in the 1990s [1]. In delayed kinetic models, we assume that the consumption of the reactants is immediate, while the product formation for any delayed reaction starts after a given constant time specific to the reaction. Recently, the notion of complex balance and the corresponding stability properties have been generalized for delayed reaction networks [2]. In this lecture, we address the kinetic realization problem for delayed models, where the task is to construct a reaction graph for a set of polynomial delayed differential equations. We define the reaction graph for delayed models similarly to [2,3]. Then, necessary and sufficient conditions are given for the kinetic realizability of nonnegative delayed polynomial models, and it is shown that such models can be rewritten into a form where the complex composition and the structure of the reaction graph explicitly appear. Using this representation, we give a systematic method to compute the so-called canonical reaction graph of delayed kinetic models, which is known from [4] in the classical non-delayed case.

[1] Roussel, M.R. (1996). The use of delay differential equations in chemical kinetics. The Journal of Physical Chemistry, 100(20), 8323-8330.

[2] Lipták, Gy., Hangos, K. M., Pituk, M. and Szederkényi, G (2018).  Semistability of complex balanced kinetic systems with arbitrary time delays. Systems & Control Letters, 114, 38-43.

[3] Mincheva, M. and Roussel, M.R. (2007). Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays. Journal of Mathematical Biology, 55, 87-104.

[4] Hárs, V. and Tóth, J. (1981). On the inverse problem of reaction kinetics. In M. Farkas and L. Hatvani (eds.), Qualitative Theory of Differential Equations, volume 30 of Coll. Math. Soc. J. Bolyai, 363-379. North-Holland, Amsterdam.  

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Az előadás a Formális reakciókinetikai szeminárium előadása.